L(s) = 1 | + (−1.19 + 0.756i)2-s + (0.856 − 1.80i)4-s − 4.10i·5-s + 7-s + (0.343 + 2.80i)8-s + (3.10 + 4.90i)10-s + 2.67i·11-s − 3.02i·13-s + (−1.19 + 0.756i)14-s + (−2.53 − 3.09i)16-s + 5.12·17-s − 2.78i·19-s + (−7.41 − 3.51i)20-s + (−2.02 − 3.19i)22-s − 7.12·23-s + ⋯ |
L(s) = 1 | + (−0.845 + 0.534i)2-s + (0.428 − 0.903i)4-s − 1.83i·5-s + 0.377·7-s + (0.121 + 0.992i)8-s + (0.981 + 1.55i)10-s + 0.807i·11-s − 0.838i·13-s + (−0.319 + 0.202i)14-s + (−0.633 − 0.773i)16-s + 1.24·17-s − 0.637i·19-s + (−1.65 − 0.785i)20-s + (−0.431 − 0.682i)22-s − 1.48·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.121 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.121 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.652325 - 0.577334i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.652325 - 0.577334i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.19 - 0.756i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + 4.10iT - 5T^{2} \) |
| 11 | \( 1 - 2.67iT - 11T^{2} \) |
| 13 | \( 1 + 3.02iT - 13T^{2} \) |
| 17 | \( 1 - 5.12T + 17T^{2} \) |
| 19 | \( 1 + 2.78iT - 19T^{2} \) |
| 23 | \( 1 + 7.12T + 23T^{2} \) |
| 29 | \( 1 + 8.83iT - 29T^{2} \) |
| 31 | \( 1 + 1.42T + 31T^{2} \) |
| 37 | \( 1 + 1.42iT - 37T^{2} \) |
| 41 | \( 1 + 5.12T + 41T^{2} \) |
| 43 | \( 1 + 2.39iT - 43T^{2} \) |
| 47 | \( 1 - 9.56T + 47T^{2} \) |
| 53 | \( 1 + 2.78iT - 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 - 5.17iT - 61T^{2} \) |
| 67 | \( 1 - 0.244iT - 67T^{2} \) |
| 71 | \( 1 - 4.27T + 71T^{2} \) |
| 73 | \( 1 + 4.15T + 73T^{2} \) |
| 79 | \( 1 - 6.25T + 79T^{2} \) |
| 83 | \( 1 + 9.35iT - 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 - 6.69T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.28533231332801297061159328885, −9.718559286654454627348350139245, −8.832118761155878983713157536174, −8.044330232177693814165179140587, −7.51723939462532092463076526604, −5.90337893711496655524306614090, −5.26249218496061799532573904290, −4.27696268260208546521098781286, −1.98894732046040587315788481101, −0.69810166096243706579541683139,
1.82482043756576074841121446796, 3.08878082193135043807338288817, 3.82460985056023563629016957962, 5.86126396549973120084691332221, 6.79394240745581040817273494402, 7.59509921348481940258036280068, 8.390563780803850017617230491217, 9.618406022181595106139587239462, 10.35149481568703580864662485693, 10.95623409057398416159455216725